@c DO NOT EDIT!  Generated automatically by munge-texi.pl.

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@node Optimization
@chapter Optimization

Octave comes with support for solving various kinds of optimization
problems.  Specifically Octave can solve problems in Linear Programming,
Quadratic Programming, Nonlinear Programming, and Linear Least Squares
Minimization.

@menu
* Linear Programming::
* Quadratic Programming::
* Nonlinear Programming::
* Linear Least Squares::
@end menu

@c @cindex linear programming
@cindex quadratic programming
@cindex nonlinear programming
@cindex optimization
@cindex LP
@cindex QP
@cindex NLP

@node Linear Programming
@section Linear Programming

Octave can solve Linear Programming problems using the @code{glpk}
function.  That is, Octave can solve

@tex
$$
  \min_x c^T x
$$
@end tex
@ifnottex

@example
min C'*x
@end example

@end ifnottex
subject to the linear constraints
@tex
$Ax = b$ where $x \geq 0$.
@end tex
@ifnottex
@math{A*x = b} where @math{x @geq{} 0}.
@end ifnottex

@noindent
The @code{glpk} function also supports variations of this problem.

@c glpk scripts/optimization/glpk.m
@anchor{XREFglpk}
@deftypefn {} {[@var{xopt}, @var{fmin}, @var{errnum}, @var{extra}] =} glpk (@var{c}, @var{A}, @var{b}, @var{lb}, @var{ub}, @var{ctype}, @var{vartype}, @var{sense}, @var{param})
Solve a linear program using the GNU @sc{glpk} library.

Given three arguments, @code{glpk} solves the following standard LP:
@tex
$$
  \min_x C^T x
$$
@end tex
@ifnottex

@example
min C'*x
@end example

@end ifnottex
subject to
@tex
$$
  Ax = b \qquad x \geq 0
$$
@end tex
@ifnottex

@example
@group
A*x  = b
  x >= 0
@end group
@end example

@end ifnottex
but may also solve problems of the form
@tex
$$
  [ \min_x | \max_x ] C^T x
$$
@end tex
@ifnottex

@example
[ min | max ] C'*x
@end example

@end ifnottex
subject to
@tex
$$
 Ax [ = | \leq | \geq ] b \qquad LB \leq x \leq UB
$$
@end tex
@ifnottex

@example
@group
A*x [ "=" | "<=" | ">=" ] b
  x >= LB
  x <= UB
@end group
@end example

@end ifnottex

Input arguments:

@table @var
@item c
A column array containing the objective function coefficients.

@item A
A matrix containing the constraints coefficients.

@item b
A column array containing the right-hand side value for each constraint in
the constraint matrix.

@item lb
An array containing the lower bound on each of the variables.  If @var{lb}
is not supplied, the default lower bound for the variables is zero.

@item ub
An array containing the upper bound on each of the variables.  If @var{ub}
is not supplied, the default upper bound is assumed to be infinite.

@item ctype
An array of characters containing the sense of each constraint in the
constraint matrix.  Each element of the array may be one of the following
values

@table @asis
@item @qcode{"F"}
A free (unbounded) constraint (the constraint is ignored).

@item @qcode{"U"}
An inequality constraint with an upper bound (@code{A(i,:)*x <= b(i)}).

@item @qcode{"S"}
An equality constraint (@code{A(i,:)*x = b(i)}).

@item @qcode{"L"}
An inequality with a lower bound (@code{A(i,:)*x >= b(i)}).

@item @qcode{"D"}
An inequality constraint with both upper and lower bounds
(@code{A(i,:)*x >= -b(i)}) @emph{and} (@code{A(i,:)*x <= b(i)}).
@end table

@item vartype
A column array containing the types of the variables.

@table @asis
@item @qcode{"C"}
A continuous variable.

@item @qcode{"I"}
An integer variable.
@end table

@item sense
If @var{sense} is 1, the problem is a minimization.  If @var{sense} is -1,
the problem is a maximization.  The default value is 1.

@item param
A structure containing the following parameters used to define the
behavior of solver.  Missing elements in the structure take on default
values, so you only need to set the elements that you wish to change from
the default.

Integer parameters:

@table @code
@item msglev (default: 1)
Level of messages output by solver routines:

@table @asis
@item 0 (@w{@code{GLP_MSG_OFF}})
No output.

@item 1 (@w{@code{GLP_MSG_ERR}})
Error and warning messages only.

@item 2 (@w{@code{GLP_MSG_ON}})
Normal output.

@item 3 (@w{@code{GLP_MSG_ALL}})
Full output (includes informational messages).
@end table

@item scale (default: 16)
Scaling option.  The values can be combined with the bitwise OR operator and
may be the following:

@table @asis
@item 1 (@w{@code{GLP_SF_GM}})
Geometric mean scaling.

@item 16 (@w{@code{GLP_SF_EQ}})
Equilibration scaling.

@item 32 (@w{@code{GLP_SF_2N}})
Round scale factors to power of two.

@item 64 (@w{@code{GLP_SF_SKIP}})
Skip if problem is well scaled.
@end table

Alternatively, a value of 128 (@w{@env{GLP_SF_AUTO}}) may be also
specified, in which case the routine chooses the scaling options
automatically.

@item dual (default: 1)
Simplex method option:

@table @asis
@item 1 (@w{@code{GLP_PRIMAL}})
Use two-phase primal simplex.

@item 2 (@w{@code{GLP_DUALP}})
Use two-phase dual simplex, and if it fails, switch to the primal simplex.

@item 3 (@w{@code{GLP_DUAL}})
Use two-phase dual simplex.
@end table

@item price (default: 34)
Pricing option (for both primal and dual simplex):

@table @asis
@item 17 (@w{@code{GLP_PT_STD}})
Textbook pricing.

@item 34 (@w{@code{GLP_PT_PSE}})
Steepest edge pricing.
@end table

@item itlim (default: intmax)
Simplex iterations limit.  It is decreased by one each time when one simplex
iteration has been performed, and reaching zero value signals the solver to
stop the search.

@item outfrq (default: 200)
Output frequency, in iterations.  This parameter specifies how frequently
the solver sends information about the solution to the standard output.

@item branch (default: 4)
Branching technique option (for MIP only):

@table @asis
@item 1 (@w{@code{GLP_BR_FFV}})
First fractional variable.

@item 2 (@w{@code{GLP_BR_LFV}})
Last fractional variable.

@item 3 (@w{@code{GLP_BR_MFV}})
Most fractional variable.

@item 4 (@w{@code{GLP_BR_DTH}})
Heuristic by @nospell{Driebeck and Tomlin}.

@item 5 (@w{@code{GLP_BR_PCH}})
Hybrid @nospell{pseudocost} heuristic.
@end table

@item btrack (default: 4)
Backtracking technique option (for MIP only):

@table @asis
@item 1 (@w{@code{GLP_BT_DFS}})
Depth first search.

@item 2 (@w{@code{GLP_BT_BFS}})
Breadth first search.

@item 3 (@w{@code{GLP_BT_BLB}})
Best local bound.

@item 4 (@w{@code{GLP_BT_BPH}})
Best projection heuristic.
@end table

@item presol (default: 1)
If this flag is set, the simplex solver uses the built-in LP presolver.
Otherwise the LP presolver is not used.

@item lpsolver (default: 1)
Select which solver to use.  If the problem is a MIP problem this flag
will be ignored.

@table @asis
@item 1
Revised simplex method.

@item 2
Interior point method.
@end table

@item rtest (default: 34)
Ratio test technique:

@table @asis
@item 17 (@w{@code{GLP_RT_STD}})
Standard ("textbook").

@item 34 (@w{@code{GLP_RT_HAR}})
Harris' two-pass ratio test.
@end table

@item tmlim (default: intmax)
Searching time limit, in milliseconds.

@item outdly (default: 0)
Output delay, in seconds.  This parameter specifies how long the solver
should delay sending information about the solution to the standard output.

@item save (default: 0)
If this parameter is nonzero, save a copy of the problem in @nospell{CPLEX}
LP format to the file @file{"outpb.lp"}.  There is currently no way to
change the name of the output file.
@end table

Real parameters:

@table @code
@item tolbnd (default: 1e-7)
Relative tolerance used to check if the current basic solution is primal
feasible.  It is not recommended that you change this parameter unless you
have a detailed understanding of its purpose.

@item toldj (default: 1e-7)
Absolute tolerance used to check if the current basic solution is dual
feasible.  It is not recommended that you change this parameter unless you
have a detailed understanding of its purpose.

@item tolpiv (default: 1e-10)
Relative tolerance used to choose eligible pivotal elements of the simplex
table.  It is not recommended that you change this parameter unless you have
a detailed understanding of its purpose.

@item objll (default: -DBL_MAX)
Lower limit of the objective function.  If the objective function reaches
this limit and continues decreasing, the solver stops the search.  This
parameter is used in the dual simplex method only.

@item objul (default: +DBL_MAX)
Upper limit of the objective function.  If the objective function reaches
this limit and continues increasing, the solver stops the search.  This
parameter is used in the dual simplex only.

@item tolint (default: 1e-5)
Relative tolerance used to check if the current basic solution is integer
feasible.  It is not recommended that you change this parameter unless you
have a detailed understanding of its purpose.

@item tolobj (default: 1e-7)
Relative tolerance used to check if the value of the objective function is
not better than in the best known integer feasible solution.  It is not
recommended that you change this parameter unless you have a detailed
understanding of its purpose.
@end table
@end table

Output values:

@table @var
@item xopt
The optimizer (the value of the decision variables at the optimum).

@item fopt
The optimum value of the objective function.

@item errnum
Error code.

@table @asis
@item 0
No error.

@item 1 (@w{@code{GLP_EBADB}})
Invalid basis.

@item 2 (@w{@code{GLP_ESING}})
Singular matrix.

@item 3 (@w{@code{GLP_ECOND}})
Ill-conditioned matrix.

@item 4 (@w{@code{GLP_EBOUND}})
Invalid bounds.

@item 5 (@w{@code{GLP_EFAIL}})
Solver failed.

@item 6 (@w{@code{GLP_EOBJLL}})
Objective function lower limit reached.

@item 7 (@w{@code{GLP_EOBJUL}})
Objective function upper limit reached.

@item 8 (@w{@code{GLP_EITLIM}})
Iterations limit exhausted.

@item 9 (@w{@code{GLP_ETMLIM}})
Time limit exhausted.

@item 10 (@w{@code{GLP_ENOPFS}})
No primal feasible solution.

@item 11 (@w{@code{GLP_ENODFS}})
No dual feasible solution.

@item 12 (@w{@code{GLP_EROOT}})
Root LP optimum not provided.

@item 13 (@w{@code{GLP_ESTOP}})
Search terminated by application.

@item 14 (@w{@code{GLP_EMIPGAP}})
Relative MIP gap tolerance reached.

@item 15 (@w{@code{GLP_ENOFEAS}})
No primal/dual feasible solution.

@item 16 (@w{@code{GLP_ENOCVG}})
No convergence.

@item 17 (@w{@code{GLP_EINSTAB}})
Numerical instability.

@item 18 (@w{@code{GLP_EDATA}})
Invalid data.

@item 19 (@w{@code{GLP_ERANGE}})
Result out of range.
@end table

@item extra
A data structure containing the following fields:

@table @code
@item lambda
Dual variables.

@item redcosts
Reduced Costs.

@item time
Time (in seconds) used for solving LP/MIP problem.

@item status
Status of the optimization.

@table @asis
@item 1 (@w{@code{GLP_UNDEF}})
Solution status is undefined.

@item 2 (@w{@code{GLP_FEAS}})
Solution is feasible.

@item 3 (@w{@code{GLP_INFEAS}})
Solution is infeasible.

@item 4 (@w{@code{GLP_NOFEAS}})
Problem has no feasible solution.

@item 5 (@w{@code{GLP_OPT}})
Solution is optimal.

@item 6 (@w{@code{GLP_UNBND}})
Problem has no unbounded solution.
@end table
@end table
@end table

Example:

@example
@group
c = [10, 6, 4]';
A = [ 1, 1, 1;
     10, 4, 5;
      2, 2, 6];
b = [100, 600, 300]';
lb = [0, 0, 0]';
ub = [];
ctype = "UUU";
vartype = "CCC";
s = -1;

param.msglev = 1;
param.itlim = 100;

[xmin, fmin, status, extra] = ...
   glpk (c, A, b, lb, ub, ctype, vartype, s, param);
@end group
@end example
@end deftypefn


@node Quadratic Programming
@section Quadratic Programming

Octave can also solve Quadratic Programming problems, this is
@tex
$$
 \min_x {1 \over 2} x^T H x + x^T q
$$
@end tex
@ifnottex

@example
min 0.5 x'*H*x + x'*q
@end example

@end ifnottex
subject to
@tex
$$
 A x = b \qquad lb \leq x \leq ub \qquad A_{lb} \leq A_{in} x \leq A_{ub}
$$
@end tex
@ifnottex

@example
@group
     A*x = b
     lb <= x <= ub
     A_lb <= A_in*x <= A_ub
@end group
@end example

@end ifnottex

@c qp scripts/optimization/qp.m
@anchor{XREFqp}
@deftypefn  {} {[@var{x}, @var{obj}, @var{info}, @var{lambda}] =} qp (@var{x0}, @var{H})
@deftypefnx {} {[@var{x}, @var{obj}, @var{info}, @var{lambda}] =} qp (@var{x0}, @var{H}, @var{q})
@deftypefnx {} {[@var{x}, @var{obj}, @var{info}, @var{lambda}] =} qp (@var{x0}, @var{H}, @var{q}, @var{A}, @var{b})
@deftypefnx {} {[@var{x}, @var{obj}, @var{info}, @var{lambda}] =} qp (@var{x0}, @var{H}, @var{q}, @var{A}, @var{b}, @var{lb}, @var{ub})
@deftypefnx {} {[@var{x}, @var{obj}, @var{info}, @var{lambda}] =} qp (@var{x0}, @var{H}, @var{q}, @var{A}, @var{b}, @var{lb}, @var{ub}, @var{A_lb}, @var{A_in}, @var{A_ub})
@deftypefnx {} {[@var{x}, @var{obj}, @var{info}, @var{lambda}] =} qp (@dots{}, @var{options})
Solve a quadratic program (QP).

Solve the quadratic program defined by
@tex
$$
 \min_x {1 \over 2} x^T H x + x^T q
$$
@end tex
@ifnottex

@example
@group
min 0.5 x'*H*x + x'*q
 x
@end group
@end example

@end ifnottex
subject to
@tex
$$
 A x = b \qquad lb \leq x \leq ub \qquad A_{lb} \leq A_{in} x \leq A_{ub}
$$
@end tex
@ifnottex

@example
@group
A*x = b
lb <= x <= ub
A_lb <= A_in*x <= A_ub
@end group
@end example

@end ifnottex
@noindent
using a null-space active-set method.

Any bound (@var{A}, @var{b}, @var{lb}, @var{ub}, @var{A_in}, @var{A_lb},
@var{A_ub}) may be set to the empty matrix (@code{[]}) if not present.  The
constraints @var{A} and @var{A_in} are matrices with each row representing
a single constraint.  The other bounds are scalars or vectors depending on
the number of constraints.  The algorithm is faster if the initial guess is
feasible.

@var{options} is a structure specifying additional parameters which
control the algorithm.  Currently, @code{qp} recognizes these options:
@qcode{"MaxIter"}, @qcode{"TolX"}.

@qcode{"MaxIter"} proscribes the maximum number of algorithm iterations
before optimization is halted.  The default value is 200.
The value must be a positive integer.

@qcode{"TolX"} specifies the termination tolerance for the unknown variables
@var{x}.  The default is @code{sqrt (eps)} or approximately 1e-8.

On return, @var{x} is the location of the minimum and @var{fval} contains
the value of the objective function at @var{x}.

@table @var
@item info
Structure containing run-time information about the algorithm.  The
following fields are defined:

@table @code
@item solveiter
The number of iterations required to find the solution.

@item info
An integer indicating the status of the solution.

@table @asis
@item 0
The problem is feasible and convex.  Global solution found.

@item 1
The problem is not convex.  Local solution found.

@item 2
The problem is not convex and unbounded.

@item 3
Maximum number of iterations reached.

@item 6
The problem is infeasible.
@end table
@end table
@end table
@xseealso{@ref{XREFsqp,,sqp}}
@end deftypefn


@c pqpnonneg scripts/optimization/pqpnonneg.m
@anchor{XREFpqpnonneg}
@deftypefn  {} {@var{x} =} pqpnonneg (@var{c}, @var{d})
@deftypefnx {} {@var{x} =} pqpnonneg (@var{c}, @var{d}, @var{x0})
@deftypefnx {} {@var{x} =} pqpnonneg (@var{c}, @var{d}, @var{x0}, @var{options})
@deftypefnx {} {[@var{x}, @var{minval}] =} pqpnonneg (@dots{})
@deftypefnx {} {[@var{x}, @var{minval}, @var{exitflag}] =} pqpnonneg (@dots{})
@deftypefnx {} {[@var{x}, @var{minval}, @var{exitflag}, @var{output}] =} pqpnonneg (@dots{})
@deftypefnx {} {[@var{x}, @var{minval}, @var{exitflag}, @var{output}, @var{lambda}] =} pqpnonneg (@dots{})

Minimize @code{1/2*@var{x}'*@var{c}*@var{x} + @var{d}'*@var{x}} subject to
@code{@var{x} >= 0}.

@var{c} and @var{d} must be real matrices, and @var{c} must be symmetric and
positive definite.

@var{x0} is an optional initial guess for the solution @var{x}.

@var{options} is an options structure to change the behavior of the
algorithm (@pxref{XREFoptimset,,@code{optimset}}).  @code{pqpnonneg}
recognizes one option: @qcode{"MaxIter"}.

Outputs:

@table @var

@item x
The solution matrix

@item minval
The minimum attained model value,
@code{1/2*@var{xmin}'*@var{c}*@var{xmin} + @var{d}'*@var{xmin}}

@item exitflag
An indicator of convergence.  0 indicates that the iteration count was
exceeded, and therefore convergence was not reached; >0 indicates that the
algorithm converged.  (The algorithm is stable and will converge given
enough iterations.)

@item output
A structure with two fields:

@itemize @bullet
@item @qcode{"algorithm"}: The algorithm used (@nospell{@qcode{"nnls"}})

@item @qcode{"iterations"}: The number of iterations taken.
@end itemize

@item lambda
@c FIXME: Something is output from the function, but what is it?
Undocumented output
@end table
@xseealso{@ref{XREFlsqnonneg,,lsqnonneg}, @ref{XREFqp,,qp}, @ref{XREFoptimset,,optimset}}
@end deftypefn


@node Nonlinear Programming
@section Nonlinear Programming

Octave can also perform general nonlinear minimization using a
successive quadratic programming solver.

@c sqp scripts/optimization/sqp.m
@anchor{XREFsqp}
@deftypefn  {} {[@var{x}, @var{obj}, @var{info}, @var{iter}, @var{nf}, @var{lambda}] =} sqp (@var{x0}, @var{phi})
@deftypefnx {} {[@dots{}] =} sqp (@var{x0}, @var{phi}, @var{g})
@deftypefnx {} {[@dots{}] =} sqp (@var{x0}, @var{phi}, @var{g}, @var{h})
@deftypefnx {} {[@dots{}] =} sqp (@var{x0}, @var{phi}, @var{g}, @var{h}, @var{lb}, @var{ub})
@deftypefnx {} {[@dots{}] =} sqp (@var{x0}, @var{phi}, @var{g}, @var{h}, @var{lb}, @var{ub}, @var{maxiter})
@deftypefnx {} {[@dots{}] =} sqp (@var{x0}, @var{phi}, @var{g}, @var{h}, @var{lb}, @var{ub}, @var{maxiter}, @var{tolerance})
Minimize an objective function using sequential quadratic programming (SQP).

Solve the nonlinear program
@tex
$$
\min_x \phi (x)
$$
@end tex
@ifnottex

@example
@group
min phi (x)
 x
@end group
@end example

@end ifnottex
subject to
@tex
$$
 g(x) = 0 \qquad h(x) \geq 0 \qquad lb \leq x \leq ub
$$
@end tex
@ifnottex

@example
@group
g(x)  = 0
h(x) >= 0
lb <= x <= ub
@end group
@end example

@end ifnottex
@noindent
using a sequential quadratic programming method.

The first argument is the initial guess for the vector @var{x0}.

The second argument is a function handle pointing to the objective function
@var{phi}.  The objective function must accept one vector argument and
return a scalar.

The second argument may also be a 2- or 3-element cell array of function
handles.  The first element should point to the objective function, the
second should point to a function that computes the gradient of the
objective function, and the third should point to a function that computes
the Hessian of the objective function.  If the gradient function is not
supplied, the gradient is computed by finite differences.  If the Hessian
function is not supplied, a BFGS update formula is used to approximate the
Hessian.

When supplied, the gradient function @code{@var{phi}@{2@}} must accept one
vector argument and return a vector.  When supplied, the Hessian function
@code{@var{phi}@{3@}} must accept one vector argument and return a matrix.

The third and fourth arguments @var{g} and @var{h} are function handles
pointing to functions that compute the equality constraints and the
inequality constraints, respectively.  If the problem does not have
equality (or inequality) constraints, then use an empty matrix ([]) for
@var{g} (or @var{h}).  When supplied, these equality and inequality
constraint functions must accept one vector argument and return a vector.

The third and fourth arguments may also be 2-element cell arrays of
function handles.  The first element should point to the constraint
function and the second should point to a function that computes the
gradient of the constraint function:
@tex
$$
 \Bigg( {\partial f(x) \over \partial x_1},
        {\partial f(x) \over \partial x_2}, \ldots,
        {\partial f(x) \over \partial x_N} \Bigg)^T
$$
@end tex
@ifnottex

@example
@group
            [ d f(x)   d f(x)        d f(x) ]
transpose ( [ ------   -----   ...   ------ ] )
            [  dx_1     dx_2          dx_N  ]
@end group
@end example

@end ifnottex
The fifth and sixth arguments, @var{lb} and @var{ub}, contain lower and
upper bounds on @var{x}.  These must be consistent with the equality and
inequality constraints @var{g} and @var{h}.  If the arguments are vectors
then @var{x}(i) is bound by @var{lb}(i) and @var{ub}(i).  A bound can also
be a scalar in which case all elements of @var{x} will share the same
bound.  If only one bound (lb, ub) is specified then the other will
default to (-@var{realmax}, +@var{realmax}).

The seventh argument @var{maxiter} specifies the maximum number of
iterations.  The default value is 100.

The eighth argument @var{tolerance} specifies the tolerance for the stopping
criteria.  The default value is @code{sqrt (eps)}.

The value returned in @var{info} may be one of the following:

@table @asis
@item 101
The algorithm terminated normally.
All constraints meet the specified tolerance.

@item 102
The BFGS update failed.

@item 103
The maximum number of iterations was reached.

@item 104
The stepsize has become too small, i.e.,
@tex
$\Delta x,$
@end tex
@ifnottex
delta @var{x},
@end ifnottex
is less than @code{@var{tol} * norm (x)}.
@end table

An example of calling @code{sqp}:

@example
function r = g (x)
  r = [ sumsq(x)-10;
        x(2)*x(3)-5*x(4)*x(5);
        x(1)^3+x(2)^3+1 ];
endfunction

function obj = phi (x)
  obj = exp (prod (x)) - 0.5*(x(1)^3+x(2)^3+1)^2;
endfunction

x0 = [-1.8; 1.7; 1.9; -0.8; -0.8];

[x, obj, info, iter, nf, lambda] = sqp (x0, @@phi, @@g, [])

x =

  -1.71714
   1.59571
   1.82725
  -0.76364
  -0.76364

obj = 0.053950
info = 101
iter = 8
nf = 10
lambda =

  -0.0401627
   0.0379578
  -0.0052227
@end example

@xseealso{@ref{XREFqp,,qp}}
@end deftypefn


@node Linear Least Squares
@section Linear Least Squares

Octave also supports linear least squares minimization.  That is,
Octave can find the parameter @math{b} such that the model
@tex
$y = xb$
@end tex
@ifnottex
@math{y = x*b}
@end ifnottex
fits data @math{(x,y)} as well as possible, assuming zero-mean
Gaussian noise.  If the noise is assumed to be isotropic the problem
can be solved using the @samp{\} or @samp{/} operators, or the @code{ols}
function.  In the general case where the noise is assumed to be anisotropic
the @code{gls} is needed.

@c ols scripts/linear-algebra/ols.m
@anchor{XREFols}
@deftypefn {} {[@var{beta}, @var{sigma}, @var{r}] =} ols (@var{y}, @var{x})
Ordinary least squares (OLS) estimation.

OLS applies to the multivariate model
@tex
$@var{y} = @var{x}\,@var{b} + @var{e}$
@end tex
@ifnottex
@w{@math{@var{y} = @var{x}*@var{b} + @var{e}}}
@end ifnottex
where
@tex
$@var{y}$ is a $t \times p$ matrix, $@var{x}$ is a $t \times k$ matrix,
$@var{b}$ is a $k \times p$ matrix, and $@var{e}$ is a $t \times p$ matrix.
@end tex
@ifnottex
@math{@var{y}} is a @math{t}-by-@math{p} matrix, @math{@var{x}} is a
@math{t}-by-@math{k} matrix, @var{b} is a @math{k}-by-@math{p} matrix, and
@var{e} is a @math{t}-by-@math{p} matrix.
@end ifnottex

Each row of @var{y} is a @math{p}-variate observation in which each column
represents a variable.  Likewise, the rows of @var{x} represent
@math{k}-variate observations or possibly designed values.  Furthermore,
the collection of observations @var{x} must be of adequate rank, @math{k},
otherwise @var{b} cannot be uniquely estimated.

The observation errors, @var{e}, are assumed to originate from an
underlying @math{p}-variate distribution with zero mean and
@math{p}-by-@math{p} covariance matrix @var{S}, both constant conditioned
on @var{x}.  Furthermore, the matrix @var{S} is constant with respect to
each observation such that
@tex
$\bar{@var{e}} = 0$ and cov(vec(@var{e})) =  kron(@var{s},@var{I}).
@end tex
@ifnottex
@code{mean (@var{e}) = 0} and
@code{cov (vec (@var{e})) = kron (@var{s}, @var{I})}.
@end ifnottex
(For cases
that don't meet this criteria, such as autocorrelated errors, see
generalized least squares, gls, for more efficient estimations.)

The return values @var{beta}, @var{sigma}, and @var{r} are defined as
follows.

@table @var
@item beta
The OLS estimator for matrix @var{b}.
@tex
@var{beta} is calculated directly via $(@var{x}^T@var{x})^{-1} @var{x}^T
@var{y}$ if the matrix $@var{x}^T@var{x}$ is of full rank.
@end tex
@ifnottex
@var{beta} is calculated directly via
@code{inv (@var{x}'*@var{x}) * @var{x}' * @var{y}} if the matrix
@code{@var{x}'*@var{x}} is of full rank.
@end ifnottex
Otherwise, @code{@var{beta} = pinv (@var{x}) * @var{y}} where
@code{pinv (@var{x})} denotes the pseudoinverse of @var{x}.

@item sigma
The OLS estimator for the matrix @var{s},

@example
@group
@var{sigma} = (@var{y}-@var{x}*@var{beta})' * (@var{y}-@var{x}*@var{beta}) / (@math{t}-rank(@var{x}))
@end group
@end example

@item r
The matrix of OLS residuals, @code{@var{r} = @var{y} - @var{x}*@var{beta}}.
@end table
@xseealso{@ref{XREFgls,,gls}, @ref{XREFpinv,,pinv}}
@end deftypefn


@c gls scripts/linear-algebra/gls.m
@anchor{XREFgls}
@deftypefn {} {[@var{beta}, @var{v}, @var{r}] =} gls (@var{y}, @var{x}, @var{o})
Generalized least squares (GLS) model.

Perform a generalized least squares estimation for the multivariate model
@tex
$@var{y} = @var{x}\,@var{b} + @var{e}$
@end tex
@ifnottex
@w{@math{@var{y} = @var{x}*@var{B} + @var{E}}}
@end ifnottex
where
@tex
$@var{y}$ is a $t \times p$ matrix, $@var{x}$ is a $t \times k$ matrix,
$@var{b}$ is a $k \times p$ matrix and $@var{e}$ is a $t \times p$ matrix.
@end tex
@ifnottex
@var{y} is a @math{t}-by-@math{p} matrix, @var{x} is a
@math{t}-by-@math{k} matrix, @var{b} is a @math{k}-by-@math{p} matrix
and @var{e} is a @math{t}-by-@math{p} matrix.
@end ifnottex

@noindent
Each row of @var{y} is a @math{p}-variate observation in which each column
represents a variable.  Likewise, the rows of @var{x} represent
@math{k}-variate observations or possibly designed values.  Furthermore,
the collection of observations @var{x} must be of adequate rank, @math{k},
otherwise @var{b} cannot be uniquely estimated.

The observation errors, @var{e}, are assumed to originate from an
underlying @math{p}-variate distribution with zero mean but possibly
heteroscedastic observations.  That is, in general,
@tex
$\bar{@var{e}} = 0$ and cov(vec(@var{e})) = $s^2@var{o}$
@end tex
@ifnottex
@code{@math{mean (@var{e}) = 0}} and
@code{@math{cov (vec (@var{e})) = (@math{s}^2)*@var{o}}}
@end ifnottex
in which @math{s} is a scalar and @var{o} is a
@tex
@math{t \, p \times t \, p}
@end tex
@ifnottex
@math{t*p}-by-@math{t*p}
@end ifnottex
matrix.

The return values @var{beta}, @var{v}, and @var{r} are
defined as follows.

@table @var
@item beta
The GLS estimator for matrix @var{b}.

@item v
The GLS estimator for scalar @math{s^2}.

@item r
The matrix of GLS residuals, @math{@var{r} = @var{y} - @var{x}*@var{beta}}.
@end table
@xseealso{@ref{XREFols,,ols}}
@end deftypefn


@c lsqnonneg scripts/optimization/lsqnonneg.m
@anchor{XREFlsqnonneg}
@deftypefn  {} {@var{x} =} lsqnonneg (@var{c}, @var{d})
@deftypefnx {} {@var{x} =} lsqnonneg (@var{c}, @var{d}, @var{x0})
@deftypefnx {} {@var{x} =} lsqnonneg (@var{c}, @var{d}, @var{x0}, @var{options})
@deftypefnx {} {[@var{x}, @var{resnorm}] =} lsqnonneg (@dots{})
@deftypefnx {} {[@var{x}, @var{resnorm}, @var{residual}] =} lsqnonneg (@dots{})
@deftypefnx {} {[@var{x}, @var{resnorm}, @var{residual}, @var{exitflag}] =} lsqnonneg (@dots{})
@deftypefnx {} {[@var{x}, @var{resnorm}, @var{residual}, @var{exitflag}, @var{output}] =} lsqnonneg (@dots{})
@deftypefnx {} {[@var{x}, @var{resnorm}, @var{residual}, @var{exitflag}, @var{output}, @var{lambda}] =} lsqnonneg (@dots{})

Minimize @code{norm (@var{c}*@var{x} - @var{d})} subject to
@code{@var{x} >= 0}.

@var{c} and @var{d} must be real matrices.

@var{x0} is an optional initial guess for the solution @var{x}.

@var{options} is an options structure to change the behavior of the
algorithm (@pxref{XREFoptimset,,@code{optimset}}).  @code{lsqnonneg}
recognizes these options: @qcode{"MaxIter"}, @qcode{"TolX"}.

Outputs:

@table @var
@item resnorm
The squared 2-norm of the residual: @code{norm (@var{c}*@var{x}-@var{d})^2}

@item residual
The residual: @code{@var{d}-@var{c}*@var{x}}

@item exitflag
An indicator of convergence.  0 indicates that the iteration count was
exceeded, and therefore convergence was not reached; >0 indicates that the
algorithm converged.  (The algorithm is stable and will converge given
enough iterations.)

@item output
A structure with two fields:

@itemize @bullet
@item @qcode{"algorithm"}: The algorithm used (@qcode{"nnls"})

@item @qcode{"iterations"}: The number of iterations taken.
@end itemize

@item lambda
@c FIXME: Something is output from the function, but what is it?
Undocumented output
@end table
@xseealso{@ref{XREFpqpnonneg,,pqpnonneg}, @ref{XREFlscov,,lscov}, @ref{XREFoptimset,,optimset}}
@end deftypefn


@c lscov scripts/linear-algebra/lscov.m
@anchor{XREFlscov}
@deftypefn  {} {@var{x} =} lscov (@var{A}, @var{b})
@deftypefnx {} {@var{x} =} lscov (@var{A}, @var{b}, @var{V})
@deftypefnx {} {@var{x} =} lscov (@var{A}, @var{b}, @var{V}, @var{alg})
@deftypefnx {} {[@var{x}, @var{stdx}, @var{mse}, @var{S}] =} lscov (@dots{})

Compute a generalized linear least squares fit.

Estimate @var{x} under the model @var{b} = @var{A}@var{x} + @var{w}, where
the noise @var{w} is assumed to follow a normal distribution with covariance
matrix @math{{\sigma^2} V}.

If the size of the coefficient matrix @var{A} is n-by-p, the size of the
vector/array of constant terms @var{b} must be n-by-k.

The optional input argument @var{V} may be an n-element vector of positive
weights (inverse variances), or an n-by-n symmetric positive semi-definite
matrix representing the covariance of @var{b}.  If @var{V} is not supplied,
the ordinary least squares solution is returned.

The @var{alg} input argument, a guidance on solution method to use, is
currently ignored.

Besides the least-squares estimate matrix @var{x} (p-by-k), the function
also returns @var{stdx} (p-by-k), the error standard deviation of estimated
@var{x}; @var{mse} (k-by-1), the estimated data error covariance scale
factors (@math{\sigma^2}); and @var{S} (p-by-p, or p-by-p-by-k if k > 1),
the error covariance of @var{x}.

Reference: @nospell{Golub and Van Loan} (1996),
@cite{Matrix Computations (3rd Ed.)}, Johns Hopkins, Section 5.6.3

@xseealso{@ref{XREFols,,ols}, @ref{XREFgls,,gls}, @ref{XREFlsqnonneg,,lsqnonneg}}
@end deftypefn


@c optimset scripts/optimization/optimset.m
@anchor{XREFoptimset}
@deftypefn  {} {} optimset ()
@deftypefnx {} {@var{options} =} optimset ()
@deftypefnx {} {@var{options} =} optimset (@var{par}, @var{val}, @dots{})
@deftypefnx {} {@var{options} =} optimset (@var{old}, @var{par}, @var{val}, @dots{})
@deftypefnx {} {@var{options} =} optimset (@var{old}, @var{new})
Create options structure for optimization functions.

When called without any input or output arguments, @code{optimset} prints
a list of all valid optimization parameters.

When called with one output and no inputs, return an options structure with
all valid option parameters initialized to @code{[]}.

When called with a list of parameter/value pairs, return an options
structure with only the named parameters initialized.

When the first input is an existing options structure @var{old}, the values
are updated from either the @var{par}/@var{val} list or from the options
structure @var{new}.

Valid parameters are:

@table @asis
@item AutoScaling

@item ComplexEqn

@item Display
Request verbose display of results from optimizations.  Values are:

@table @asis
@item @qcode{"off"} [default]
No display.

@item @qcode{"iter"}
Display intermediate results for every loop iteration.

@item @qcode{"final"}
Display the result of the final loop iteration.

@item @qcode{"notify"}
Display the result of the final loop iteration if the function has
failed to converge.
@end table

@item FinDiffType

@item FunValCheck
When enabled, display an error if the objective function returns an invalid
value (a complex number, NaN, or Inf).  Must be set to @qcode{"on"} or
@qcode{"off"} [default].  Note: the functions @code{fzero} and
@code{fminbnd} correctly handle Inf values and only complex values or NaN
will cause an error in this case.

@item GradObj
When set to @qcode{"on"}, the function to be minimized must return a
second argument which is the gradient, or first derivative, of the
function at the point @var{x}.  If set to @qcode{"off"} [default], the
gradient is computed via finite differences.

@item Jacobian
When set to @qcode{"on"}, the function to be minimized must return a
second argument which is the Jacobian, or first derivative, of the
function at the point @var{x}.  If set to @qcode{"off"} [default], the
Jacobian is computed via finite differences.

@item MaxFunEvals
Maximum number of function evaluations before optimization stops.
Must be a positive integer.

@item MaxIter
Maximum number of algorithm iterations before optimization stops.
Must be a positive integer.

@item OutputFcn
A user-defined function executed once per algorithm iteration.

@item TolFun
Termination criterion for the function output.  If the difference in the
calculated objective function between one algorithm iteration and the next
is less than @code{TolFun} the optimization stops.  Must be a positive
scalar.

@item TolX
Termination criterion for the function input.  If the difference in @var{x},
the current search point, between one algorithm iteration and the next is
less than @code{TolX} the optimization stops.  Must be a positive scalar.

@item TypicalX

@item Updating
@end table
@xseealso{@ref{XREFoptimget,,optimget}}
@end deftypefn


@c optimget scripts/optimization/optimget.m
@anchor{XREFoptimget}
@deftypefn  {} {} optimget (@var{options}, @var{parname})
@deftypefnx {} {} optimget (@var{options}, @var{parname}, @var{default})
Return the specific option @var{parname} from the optimization options
structure @var{options} created by @code{optimset}.

If @var{parname} is not defined then return @var{default} if supplied,
otherwise return an empty matrix.
@xseealso{@ref{XREFoptimset,,optimset}}
@end deftypefn

